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Generation Date: 04/01/2014
Generated By: Che ryl Shelton
Title: 10 th Grade Geometry Theorems
1.
Given: g
Prove:
h
1 and
2 are supplementary
Proof:
Statements
1. g
2.
3. m
4.
1. Given
h
1
Reasons
3
1=m 3
2 and 3 are
supplementary
2.
3. Definition of congruent angles
4. Linear Pair Postulate
5. m
3 + m 2 = 180°
5. Definition of supplementary
angles
6. m
1 + m 2 = 180°
6. Substitution property of
equality
7.
1 and 2 are
supplementary
7. Definition of supplementary
angles
Which of the following reasons completes the proof?
A. Alternate Exterior Angles Theorem
B. Alternate Interior Angles Theorem
C. Definition of congruent angles
D. Definition of linear pair
2. Proving the alternate exterior angle theorem.
Given: f
Prove:
g and line d is a transversal.
1
8
Proof:
Statements
Reasons
1. Line f is parallel to line g.
Line d is a transversal.
given
2.
1
5
If parallel lines are cut by a transversal,
then corresponding angles are congruent.
1
8
transitive property of congruence
3.
4.
Which of the following statements and reasons completes the proof?
A. 1
4 since vertical angles are congruent.
B.
5
4 since alternate interior angles are congruent.
C.
5
8 since vertical angles are congruent.
D.
4
8 since corresponding angles are congruent.
3. In the triangle below, m F = 28°, m FHG = 70°, and m E = 82°.
Determine the missing reason to prove HG
DE.
Statements
Reasons
m F = 28°, m FHG = 70°, m E = 82° given
m F + m FHG + m HGF = 180°
sum of the interior angles of a
triangle equal 180°
28° + 70° + m HGF = 180°
substitution
m HGF = 82°
property of subtraction
m HGF = m E
transitive property of equality
HG
DE
If two corresponding angles are congruent, then the third angle is opposite a set of parallel
A. lines.
If two triangles have one or more sets of congruent angles, then the lines containing the
B. bases are parallel.
If three angles of one triangle are congruent to three angles of another triangle, then the
C. two triangles are congruent and the bases are parallel.
If two lines are cut by a transversal to form congruent corresponding angles, then the two
D. lines are parallel.
4.
Given: Line f is tangent to circle C at point D,
1
2
Prove: CD
f
Proof:
Statements
Reasons
1. Line f is tangent to circle C at point D,
1
2
1. Given
2.
1 and
2 are a linear pair
2. Definition of linear
pair
3.
1 and
2 are supplementary
3. Linear Pair Postulate
4. m 1 + m 2 = 180°
4. Definition of
supplementary
angles
5. m 1 = m 2
5. Definition of
congruent angles
6. m 1 + m 1 = 180°
6. Substitution property
of equality
7. 2 · (m 1) = 180°
7. Distributive property
of equality
8. m 1 = 90°
8. Division property of
equality
9.
9. Definition of right
angle
1 is a right angle
10. CD
10.
f
Which of the following reasons completes the proof?
A. Definition of perpendicular lines and segments
B. Definition of vertical angles
C. Definition of parallel lines and segments
D. Definition of congruent angles
5.
Given:
2 and
3 and
Prove:
2
3 are a linear pair,
4 are a linear pair
4
Proof:
Statements
Reasons
1.
2 and
3 and
3 are a linear pair,
4 are a linear pair
1. Given
2.
2 and
3 and
3 are supplementary,
4 are supplementary
2.
3.
2
4
Which of the following reasons completes the proof?
A. Transitive Property
B. Congruent Complements Theorem
3. Congruent Supplements
Theorem
C. Right Angle Congruence Theorem
D. Linear Pair Postulate
6. Proving the same-side interior angle theorem.
Given: f
g and line d is a transversal.
Prove: m
3+m
5 = 180°
Proof:
Statements
Reasons
1. Line f is parallel to line g.
Line d is a transversal.
given
2. m
3+m
1 = 180°
linear pair property
3+m
5 = 180°
substitution property
3.
4. m
Which of the following statements and reasons completes the proof?
A. m 5 + m 7 = 180° since they form a linear pair.
7.
B. m
3=m
7 since corresponding angles are congruent.
C. m
D. m
1=m
1=m
4 since vertical angles are congruent.
5 since corresponding angles are congruent.
Given: Point F is the center of circle F
Prove: 2 · (m 1 + m 3) = m
Proof:
Statements
Reasons
1. Point F is the center of circle F
1. Given
2. FG, FJ, and FH are radii of circle F
3. Definition of radius
2. FG
3. Definition of radius
4.
1
FJ
FH
2,
3
4
4. Definition of isosceles
triangle
5. m 1 = m 2, m 3 = m 4
5. Definition of
congruent angles
6. m 5 = 180° - (m 1 + m 2)
m 6 = 180° - (m 3 + m 4)
6.
7. m 5 = 180° - (m 1 + m 1)
m 6 = 180° - (m 3 + m 3)
7. Substitution property
of equality
8. m 5 = 180° - 2 · (m 1)
m 6 = 180° - 2 · (m 3)
8. Distributive property
of equality
9. m 7 = 360° - (m 5 + m 6)
9. Sum of angles about
a point
10. m 7 = 360° - (180° - 2 · (m 1))
- (180° - 2 · (m 3))
10. Substitution
property of equality
11. m 7 = 2 · (m 1 + m 3)
11. Simplification
12. m
12. Definition of central
angle
= 2 · (m 1 + m 3)
Which of the following reasons completes the proof?
A. Definition of complementary angles
B. Congruent Supplements Theorem
C. Linear Pair Postulate
D. Triangle Sum Theorem
8. Given isosceles trapezoid ABCD with DF
reason to prove AFD
CEB.
CE and
Statements
ABCD is an isosceles trapezoid
C
CEB, determine the missing
Reasons
given
The base angles of an isosceles
trapezoid are congruent.
D
DF
CE
given
AB
DC
definition of isosceles trapezoid
AFD
BAF
BAF
CEB
given
AFD
CEB
transitive property
AFD
CEB
ASA
A. alternate interior angles conjecture
B. vertical angle conjecture
C. corresponding angles conjecture
D. alternate exterior angles conjecture
9.
BAF
Given: g
Prove:
h
1
3
Proof:
Statements
1. g
2.
1. Given
h
2
Reasons
3
2.
3. m 2 = m 3 3. Definition of congruent angles
4.
1
2
4. Vertical Angles Theorem
5. m 1 = m 2 5. Definition of congruent angles
6. m 1 = m 3 6. Transitive property of equality
7.
1
3
7. Definition of congruent angles
Which of the following reasons completes the proof?
A. Definition of complementary angles
B. Consecutive Interior Angles Theorem
C. Corresponding Angles Postulate
D. Definition of parallel lines
10. Given point D is on the bisector of CAB and AD is an altitude of
missing reason to prove CAD
BAD.
Statements
AD
Reasons
AD
reflexive property
AD is an altitude of CAB
AD
CDA
CAB, determine the
given
definition of altitude
CB
BDA
point D is on the bisector of
perpendicular angles are congruent
CAB
given
DAC
DAB
CAD
BAD
ASA
A. definition of isosceles triangle
B. definition of angle bisector
C. reflexive property
D. definition of an altitude
11.
Given:
1 and 2 are complementary,
3 and 4 are complementary,
1
4
Prove:
2
3
Proof:
Statements
1.
1 and 2 are complements,
3 and 4 are complements,
1
4
Reasons
1. Given
2. m 1 + m 2 = 90°
m 3 + m 4 = 90°
2. Definition of complementary
angles
3. m 1 + m 2 = m 3 + m 4
3. Transitive property of equality
4. m 1 = m 4
4.
5. m 1 + m 2 = m 3 + m 1
5. Substitution property of
equality
6. m 2 = m 3
6. Subtraction property of
equality
7.
7. Definition of congruent
angles
2
3
Which of the following reasons completes the proof?
A. Definition of congruent angles
B. Transitive Property of Angle Congruence
C. Definition of complementary angles
D. Right Angle Congruence Theorem
12. Proving the vertical angle theorem.
Given:
1 and
Prove:
1
2 are vertical angles.
2
Proof:
Statements
1.
1 and
Reasons
2 are vertical angles
2. m
m
1+m
2+m
3 = 180°
3 = 180°
3. m
1+m
3=m
4. m
1=m
2
2+m
given
linear pair property
3
Which of the following reasons completes the proof?
A. subtraction property
B. reflexive property
C. addition property
D. transitive property
13.
Given:
1 and
2 and
Prove:
1
2 are supplementary,
3 are supplementary
3
substitution property
Proof:
Statements
1.
1 and
2 and
Reasons
2 are supplements,
3 are supplements
1. Given
2. m 1 + m 2 = 180°
m 2 + m 3 = 180°
2.
3. m 1 + m 2 = m 2 + m 3
3. Transitive property of
equality
4. m 1 = m 3
4. Subtraction property of
equality
5.
5. Definition of congruent
angles
1
3
Which of the following reasons completes the proof?
A. Definition of supplementary angles
B. Definition of complementary angles
C. Transitive property of equality
D. Substitution property of equality
14.
Given:
Prove:
1
2,
1 and 2 are a linear pair
p
q
Proof:
Statements
1.
1 and
pair
2.
1 and
2 are a linear
2 are
Reasons
1. Given
2. Linear Pair Postulate
supplementary
3. m 1 + m 2 = 180°
3. Definition of supplementary
angles
4.
4. Given
1
2
5. m 1 = m 2
5. Definition of congruent
angles
6. m 1 + m 1 = 180°
6. Substitution property of
equality
7. 2 · (m 1) = 180°
7. Distributive property
8. m 1 = 90°
8. Division property of equality
9.
9. Definition of a right angle
1 is a right angle
10. p
10.
q
Which of the following reasons completes the proof?
A. Definition of parallel lines
B. Definition of perpendicular lines
C. Linear Pair Postulate
D. Congruent Supplements Theorem
15. Given m M = 23° and
obtuse.
M
P, determine the missing reason to prove
Statements
m M = 23° and
M
Reasons
P
m P = 23°
given
transitive property of equality
m M + m N + m P = 180°
23° + m N + 23° = 180°
substitution
m N = 134°
addition property of equality
N is obtuse
definition of obtuse angle
MNP is
MNP is obtuse
A.
definition of obtuse triangle
The square of the length of the third side of a triangle is equal to the sum of the squares of
the other two sides.
B. The sum of the interior angles of a triangle equals 180°.
C. Correpsonding angles sum to 180°.
D. Adjacent angles in a triangle are congruent.
16.
Given:
ABC with exterior
4
Prove: m 4 = m 1 + m 2
Proof:
Statements
Reasons
1.
ABC with exterior
2.
m 4 + m 3 = 180°
3.
m 1+m 2+m
4
Given
Linear Pair
3 = 180°
4.
5.
Triangle Sum Theorem
Substitution Property
m
4=m
1+m
2
Subtraction Property
Which of the following reasons completes the proof?
A.
m
4+m
3=m
1+m
2+m
4
B. m
4+m
3=m
1+m
2+m
3
m
1+m
3+m
4 = 180°
m
1+m
2+m
4 = 180°
C.
D.
17. Given m
proof below.
FAH = m
CBD + m
CDB, determine the missing statement in the
Statements
m
FAH = m
Reasons
CBD + m
m
DCE = m
CBD + m
m
FAH = m
DCE
CDB
given
The measure of an exterior angle of a
triangle is equal to the sum of the
CDB
measures of the remote interior angles.
transitive property of equality
converse of the alternate exterior angle
conjecture
A.
B.
m
C.
FAH = m
CBD is isosceles
DCB
D.
18. Proving the congruent supplements theorem.
Given:
1
supplementary.
3,
Prove:
4
2
1 and
2 are supplementary, and
Proof:
Statements
Reasons
3 and
4 are
1. m
1+m
2 = 180°
m
3+m
4 = 180°
2. m
1+m
2=m
3.
1
3
m
4. m
1+m
2=m
5. m
2=m
4
definition of supplementary angles
3+m
substitution property
4
3 given
1=m
1+m
4
4 subtraction property
2
Which of the following reasons completes the proof?
A. addition property
B. substitution property
C. reflexive property
D. symmetric property
19.
Given:
CB
Prove: m
BCA + m
CAB + m
ABC = 180°
Proof:
Statements
1.
2.
Reasons
Given
CB
m
DAC + m
CAB + m
BAE = 180°
Angle Addition
3.
m
DAC = m
BCA
4.
m
BAE = m
ABC
m
BCA + m
CAB + m
5.
ABC = 180°
Substitution Property
Which of the following reasons completes the proof?
Alternate Exterior Angles;
A. Alternate Exterior Angles
Alternate Exterior Angles;
B. Alternate Interior Angles
Alternate Interior Angles;
C. Alternate Exterior Angles
Alternate Interior Angles;
D. Alternate Interior Angles
20.
Given:
Prove: g
3 and
4 are a linear pair,
1 and
3 are supplementary
h
Proof:
Statements
1.
3 and
4 are a linear pair,
Reasons
1. Given
2.
1 and
3 are supplementary
3 and
4 are supplementary
3. m
1+m
3 = 180°
m
3+m
4 = 180°
2. Linear Pair Postulate
3. Definition of
supplementary angles
4. m
1+m
3=m
5. m
1=m
4
5. Subtraction property of
equality
4
6. Definition of congruent
angles
6.
7. g
1
3+m
4. Transitive property of
4
equality
7.
h
Which of the following reasons completes the proof?
A. Alternate Interior Angles Theorem
B. Alternate Exterior Angles Theorem
C. Linear Pair Postulate
D. Corresponding Angles Postulate
21. In the triangles below, NP
RP, and PM
Determine the missing reason to prove that
PQ.
MNP
Statement
Reason
NP
RP, PM
PQ
m
NPM = m
RPQ
MNP
QRP.
QRP
given
SAS
A. An angle is congruent to itself.
B. Alternate exterior angles are congruent.
C. Vertical angles are congruent.
D. Alternate interior angles are congruent.
22. Proving the alternate interior angle theorem.
Given: f
g and line d is a transversal.
Prove:
4
5
Proof:
Statements
Reasons
1. Line f is parallel to line g.
Line d is a transversal.
given
2.
1
5
If parallel lines are cut by a transversal,
then corresponding angles are congruent.
4
5
transitive property of congruence
3.
4.
Which of the following statements and reasons completes the proof?
A.
B.
C.
D.
1
8 since alternate exterior angles are congruent.
5
8 since vertical angles are congruent.
1
4 since vertical angles are congruent.
4
8 since corresponding angles are congruent.
23. In scalene triangle
WXY below,
Determine the missing reason to prove
WYA
AXB ~
BAY.
WXY.
Statements
BAY given
WYA
AB
If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
WY
XWY
XAB
XYW
XBA
X
AXB ~
A.
Reasons
X
WXY
reflexive property
If three angles of one triangle are congruent to
three angles of another triangle, then the two
triangles are similar.
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
B. If two angles are congruent to the same angle, then they are congruent.
C. If two parallel lines are cut by a transversal, then corresponding angles are congruent.
D. If two angles are vertical angles, then they are congruent.